Problem: Factor the following expression: $-5$ $x^2$ $-1$ $x+$ $4$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(4)} &=& -20 \\ {a} + {b} &=& & & {-1} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-20$ and add them together. Remember, since $-20$ is negative, one of the factors must be negative. The factors that add up to ${-1}$ will be your ${a}$ and ${b}$ When ${a}$ is ${4}$ and ${b}$ is ${-5}$ $ \begin{eqnarray} {ab} &=& ({4})({-5}) &=& -20 \\ {a} + {b} &=& {4} + {-5} &=& -1 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-5}x^2 +{4}x {-5}x +{4} $ Group the terms so that there is a common factor in each group: $ ({-5}x^2 +{4}x) + ({-5}x +{4}) $ Factor out the common factors: $ x(-5x + 4) + 1(-5x + 4) $ Notice how $(-5x + 4)$ has become a common factor. Factor this out to find the answer. $(-5x + 4)(x + 1)$